α-Specker spaces

Abstract

This paper is motivated by work on Specker spaces and a recent article of the authors on the ring of α-quotients. Here α denotes an uncountable regular cardinal or else ∞, indicating no cardinal constraint whatsoever. All spaces are compact Hausdorff, and for the most part zero-dimensional. Three strains of the “ α-Specker” condition are studied: strong, weak, and one in between which is not qualified. One of the main results characterizes these conditions, for each α and each space X, in terms of the containment of C( X) in the ring of α-quotients of S( K α X), where the latter denotes the algebra of continuous functions with finite range, defined on an appropriate cover K α X of X. “Weakly c + -Specker” is equivalent to “Specker”. The paper examines the ω 1-Specker conditions, proving that “weakly ω 1-Specker” and “ ω 1-Specker” are equivalent. In fact, it is shown that X is weakly ω 1-Specker if and only if for each f∈ C( X) there is a Baire set B with meagre complement such that f( B) is countable.